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File:Mplwp ballistic trajectories velocities.svg

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Summary

Description
English: Plot of a ballistic trajectory with air resistance. The trajectory follows the differential equation with initial conditions .

The parameters are:

  • , ,
  • The initial velocity takes the values , , , ,
The differential equation is solved numerically using Scipy odeint.
Date
Source Own work
Author Geek3
SVG development
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The SVG code is valid.
 
This plot was created with mplwp, the Matplotlib extension for Wikipedia plots.
Source code
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mplwp source code

The plot was generated with mplwp 1.0
#!/usr/bin/python
# -*- coding: utf8 -*-

import matplotlib.pyplot as plt
import matplotlib as mpl
import numpy as np
from math import *

code_website = 'http://commons.wikimedia.org/wiki/User:Geek3/mplwp'
try:
    import mplwp
except ImportError, er:
    print 'ImportError:', er
    print 'You need to download mplwp.py from', code_website
    exit(1)

name = 'mplwp_ballistic_trajectories_velocities.svg'
fig = mplwp.fig_standard(mpl)

xlim = 0,2.6; fig.gca().set_xlim(xlim)
ylim = 0,2.6*355/515.; fig.gca().set_ylim(ylim)
fig.gca().xaxis.set_major_locator(mpl.ticker.MultipleLocator(0.4))
fig.gca().yaxis.set_major_locator(mpl.ticker.MultipleLocator(0.4))

from scipy.integrate import odeint
from scipy.optimize import brentq
def ballistic(g, k, xy0, v0, alpha0, tt):
    # use a four-dimensional vector function vec = [x, y, vx, vy]
    def dif(vec, t):
        v = sqrt(vec[2]**2 + vec[3]**2)
        return [vec[2], vec[3], -k*v*vec[2], -g -k*v*vec[3]]
    
    # solve the differential equation numerically
    vec = odeint(dif, [xy0[0], xy0[1], v0*cos(alpha0), v0*sin(alpha0)], tt)
    return vec[:,0], vec[:,1] # return x(tt) and y(tt)

g = 1.0
k = 1.0
alpha0 = pi/4
for v0 in np.linspace(0, 10, 6)[1:]:
    t1 = brentq(lambda t: ballistic(g,k,[0,0],v0,alpha0,[0,t])[1][1],0.1,5)
    t = np.linspace(0, t1, 5001)
    x, y = ballistic(g, k, [0, 0], v0, alpha0, t)
    while len(y) > 1 and y[-2] <= 0.0: x = x[:-1]; y = y[:-1]
    plt.plot(x, y,
        label=ur'$v_0=\,{:.0f}$'.format(v0))

mpl.rc('legend', borderaxespad=1.0)
plt.legend(loc='upper left').get_frame().set_alpha(0.9)
plt.savefig(name)
mplwp.postprocess(name)

Licensing

I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution
This file is licensed under the Creative Commons Attribution 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
You may select the license of your choice.

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depicts

28 September 2014

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Date/TimeThumbnailDimensionsUserComment
current14:00, 28 September 2014Thumbnail for version as of 14:00, 28 September 2014600 × 400 (29 KB)Geek3ballistic_trajectories_velocities plot by Geek3

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